Elliptic curve isogeny class with LMFDB label 9680.r (Cremona label 9680w)

• Label: 9680.r
• Number of curves: $4$
• Conductor: $9680$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 9680.r have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(5\)	\(1 - T\)
\(11\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 9680.r do not have complex multiplication.

Modular form 9680.2.a.r

Isogeny matrix

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 9680.r

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
9680.r1	9680w4	\([0, 0, 0, -114587, 14925834]\)	\(22930509321/6875\)	\(49887157760000\)	\([4]\)	\(30720\)	\(1.6064\)
9680.r2	9680w3	\([0, 0, 0, -56507, -5049814]\)	\(2749884201/73205\)	\(531198455828480\)	\([2]\)	\(30720\)	\(1.6064\)
9680.r3	9680w2	\([0, 0, 0, -8107, 167706]\)	\(8120601/3025\)	\(21950349414400\)	\([2, 2]\)	\(15360\)	\(1.2599\)
9680.r4	9680w1	\([0, 0, 0, 1573, 18634]\)	\(59319/55\)	\(-399097262080\)	\([2]\)	\(7680\)	\(0.91328\)	\(\Gamma_0(N)\)-optimal